Airsoft target practice does not require full face protection. Some form of eye protection is still recommended to guard against ricochets. Using targets that trap BBs or sticky targets that are often included with the gun decreases the likelihood of ricochet.
Eye protection should be in the form of goggles or masks that fit without leaving gaps. Paintball masks work well, but there is also airsoft-specific equipment on the market now. A layer of fabric is enough to stop an airsoft BB, so goggles and a balaclava can work as well. Normal clothing is also enough to protect skin.
Shooting an Airsoft guns at an unprotected person is not safe. Some of the higher-powered guns at close range can chip a tooth or break skin. An airsoft BB fired at an eye, even from a low-powered spring gun, can cause serious damage and at point blank range is potentially lethal if you are not wearing proper equipment. Fabric gloves are also recommended.
In addition to safety equipment game rules usually prohibit shooting at close ranges, substituting a tap with a hand that counts as a "knife kill". Some players actually use soft rubber training knives for this.
An important safety consideration is the fact that airsoft guns can be very realistic and scare people who are not aware of them. If the police show up keep your hands visible and away from the gun. Better yet, put it down and take a couple of steps back.
As with most other things in life, do not do obviously stupid things and you will be OK.